Expectation and Variance

The expectation of a random variable, E(X)E(X), is the average of possible values weighted by their probabilities. Formally, it can be defined in two ways:

  1. Domain definition: E(X)=ωΩX(ω)P(ω)E(X) = \sum_{\omega \in \Omega} X(\omega) P(\omega).

  2. Range definition: E(X)=xxP(X=x)E(X) = \sum_x x P(X = x).

Expectation has nice properties of linearity: E(X+Y)=E(X)+E(Y)E(X + Y) = E(X) + E(Y) and E(aX+b)=aE(x)+bE(aX + b) = aE(x) + b.


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